Prove if $f:X\rightarrow Y$ is surjective then $|X|\ge|Y|$

Question

Prove if $f:X\rightarrow Y$ is surjective then $|X|\ge|Y|$

My work:

Let $X,Y$ sets.

Suppose $|X|\leq |Y|$ then exists a function $g:X\rightarrow Y$ injective.

Here I'm stuck, can someone help me?

Answer 1

The statement is false. Given any set $X$, the identity from $X$ onto itself is surjective. However, it's not true that $|X|>|X|$.

Answer 2

As remarked in the previous answer, you want $\geq$.

for each $y \in Y$, consider$f^{-1}(\{y\})$ and choose a unique $x$ in there. Taking the uniion of all these, there is a collection $U \subset X$ so that $f(U)=Y$, and is a biijection, so there is a bijection $g:Y \to U \subset X$